In orbital mechanics and aerospace engineering a gravitational slingshot (also known as gravity assist manoeuver or swing-by) is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate (both positively and negatively) and/or re-direct the path of spacecraft.
Over-simplified example of gravitational slingshot: the spacecraft's velocity changes by up to twice the planet's velocity.
Where does the factor of $2$ come from in this relation?
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